Sure, let's solve this step-by-step.
We have four expressions: [tex]\(a + 1\)[/tex], [tex]\(2a - 1\)[/tex], [tex]\(a + 7\)[/tex], and [tex]\(3a + 4\)[/tex], and we know that these expressions are in ascending order. The median of these four values is given as 12.
When there are four numbers, the median is the average of the middle two numbers once they are arranged in ascending order.
In this case, the median is defined as:
[tex]\[
\text{Median} = \frac{(2 \text{nd term} + 3 \text{rd term})}{2}
\][/tex]
Given that the median is 12, we can write the equation as:
[tex]\[
\frac{(2a - 1) + (a + 7)}{2} = 12
\][/tex]
Let's solve this equation step-by-step:
1. Combine the terms inside the numerator:
[tex]\[
(2a - 1) + (a + 7) = 2a - 1 + a + 7 = 3a + 6
\][/tex]
2. Now, substitute back into the median equation:
[tex]\[
\frac{3a + 6}{2} = 12
\][/tex]
3. Multiply both sides of the equation by 2 to clear the fraction:
[tex]\[
3a + 6 = 24
\][/tex]
4. Subtract 6 from both sides to isolate the term with [tex]\(a\)[/tex]:
[tex]\[
3a = 18
\][/tex]
5. Divide both sides by 3 to solve for [tex]\(a\)[/tex]:
[tex]\[
a = 6
\][/tex]
So, the value of [tex]\(a\)[/tex] is:
[tex]\[
a = 6
\][/tex]
